Universal approximation with complex-valued deep narrow neural networks

TL;DR

This paper characterizes which complex-valued neural networks with bounded width and arbitrary depth are universal approximators, identifying specific activation functions and providing width and depth bounds for approximation capabilities.

Contribution

It provides a complete characterization of universal approximation in deep narrow complex-valued neural networks, including necessary and sufficient conditions on activation functions and bounds on network width and depth.

Findings

Deep narrow complex networks are universal if activation is neither holomorphic, antiholomorphic, nor affine.

A width of 2n+2m+5 is always sufficient for universality.

A width of max{2n, 2m} is necessary in general.

Abstract

We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions $\varrho:\mathbb{C}\to \mathbb{C}$ that have the property that their associated networks are universal, i.e., are capable of approximating continuous functions to arbitrary accuracy on compact domains. Precisely, we show that deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor $\mathbb{R}$-affine. This is a much larger class of functions than in the dual setting of arbitrary width and fixed depth. Unlike in the real case, the sufficient width differs significantly depending on the considered activation function. We show that a width of $2n+2m+5$ is always sufficient and that in general a width of $max\{2n,2m\}$ is necessary. We prove, however, that a width of $n+m+3$ suffices for a rich subclass of the admissible activation functions. Here, $n$ and $m$ denote the input and output dimensions of the considered networks. Moreover, for the case of smooth and non-polyharmonic activation functions, we provide a quantitative approximation bound in terms of the depth of the considered networks.